## Differential and Integral Equations

- Differential Integral Equations
- Volume 23, Number 7/8 (2010), 703-723.

### Multiple solutions for critical elliptic systems via penalization method

Claudianor O. Alves, Giovany M. Figueiredo, and Marcelo F. Furtado

#### Abstract

We consider the system $$ \begin{cases} -\varepsilon^{2} \Delta u +W(x)u=Q_{u}(u,v)+ \frac{1}{2^*}K_u(u,v)~\text{in } \mathbb{R}^N, \\ -\varepsilon^{2} \Delta v +V(x)v=Q_{v}(u,v)+ \frac{1}{2^*}K_v(u,v)~\text{in } \mathbb{R}^N, \\ u,v \in H^{1}(\mathbb{R}^N),u(x),v(x)>0~~\text{for each } x \in \mathbb{R}^N, \end{cases} $$ where $2^*=2N/(N-2)$, $N \geq 3$, $\varepsilon>0$ is a parameter, $W$ and $V$ are positive potentials, and $Q$ and $K$ are homogeneous function with $K$ having critical growth. We relate the number of solutions to the topology of the set where $W$ and $V$ attain their minimum values. In the proof, we apply Ljusternik-Schnirelmann theory.

#### Article information

**Source**

Differential Integral Equations, Volume 23, Number 7/8 (2010), 703-723.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356019192

**Mathematical Reviews number (MathSciNet)**

MR2654266

**Zentralblatt MATH identifier**

1240.35144

**Subjects**

Primary: 35J20: Variational methods for second-order elliptic equations 35J50: Variational methods for elliptic systems 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)

#### Citation

Alves, Claudianor O.; Figueiredo, Giovany M.; Furtado, Marcelo F. Multiple solutions for critical elliptic systems via penalization method. Differential Integral Equations 23 (2010), no. 7/8, 703--723. https://projecteuclid.org/euclid.die/1356019192